# How do we determine if a statement can't be proved by mathematics alone?

How do we determine if a statement can't be proved by mathematics alone? It seems mathematics can only prove something that can be defined purely in mathematics terms, but can't prove simple statements that can't be defined or expressed in purely mathematics terms, but are there principles or statements that help us determine if a phrase expressed in natural languages can be proven mathematically? A sort of theorems or guiding principles would be nice.

• "but can't prove simple statements that can't be defined or expressed in purely mathematics terms" sounds trivial but can be tricky... From one side, if a statement is not in the context of the language of a mathematical theory, why can we imagine that that theory can prove it? Euclidean geometry does not prove facts about e.g. probability. But, in a more general setting, how can we extend the concept of proof outside mathematics? Can we prove "theorems" about history, society, etc? Sep 16, 2022 at 15:08
• As the great American philosopher Frankie Lymon asked in 1956, Why do fools fall in love? youtube.com/watch?v=2sAHiR0rkJg Sep 16, 2022 at 17:33
• Proving isolated statements is either trivial or nonsensical, it is proving large clusters of statements from few simpler ones (premises) that is of interest. And if you are asking which informal arguments (not statements) are mathematically formalizable into proofs the answer is those that can be algorithmically checked (after including context and disambiguating natural language conventions), see Azzouni, Why Do Informal Proofs Conform to Formal Norms? Sep 17, 2022 at 3:26